| L(s) = 1 | − 1.74·2-s + 1.66·3-s − 4.96·4-s − 1.47·5-s − 2.90·6-s + 22.5·7-s + 22.5·8-s − 24.2·9-s + 2.56·10-s + 11·11-s − 8.26·12-s − 39.2·14-s − 2.45·15-s + 0.288·16-s + 18.2·17-s + 42.2·18-s + 26.4·19-s + 7.30·20-s + 37.4·21-s − 19.1·22-s + 32.4·23-s + 37.6·24-s − 122.·25-s − 85.3·27-s − 111.·28-s − 156.·29-s + 4.27·30-s + ⋯ |
| L(s) = 1 | − 0.616·2-s + 0.320·3-s − 0.620·4-s − 0.131·5-s − 0.197·6-s + 1.21·7-s + 0.998·8-s − 0.897·9-s + 0.0812·10-s + 0.301·11-s − 0.198·12-s − 0.748·14-s − 0.0422·15-s + 0.00451·16-s + 0.260·17-s + 0.553·18-s + 0.319·19-s + 0.0817·20-s + 0.389·21-s − 0.185·22-s + 0.294·23-s + 0.320·24-s − 0.982·25-s − 0.608·27-s − 0.753·28-s − 1.00·29-s + 0.0260·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.74T + 8T^{2} \) |
| 3 | \( 1 - 1.66T + 27T^{2} \) |
| 5 | \( 1 + 1.47T + 125T^{2} \) |
| 7 | \( 1 - 22.5T + 343T^{2} \) |
| 17 | \( 1 - 18.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 32.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 57.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 124.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 46.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 388.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 348.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 353.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 3.65T + 2.05e5T^{2} \) |
| 61 | \( 1 - 60.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 231.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 327.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 240.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 569.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 474.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 179.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.47e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.640812424977339963687706280799, −7.72090229880997644458295629951, −7.48069745900679937774127930659, −5.92154935395985393260099216079, −5.23484580245640249773937768815, −4.33918455146677934068604808494, −3.49556648116030218070759719689, −2.15759318492497299675964759026, −1.19773545567223638783065675924, 0,
1.19773545567223638783065675924, 2.15759318492497299675964759026, 3.49556648116030218070759719689, 4.33918455146677934068604808494, 5.23484580245640249773937768815, 5.92154935395985393260099216079, 7.48069745900679937774127930659, 7.72090229880997644458295629951, 8.640812424977339963687706280799
